Normalized defining polynomial
\( x^{8} + 41x^{6} + 123x^{4} - 6847x^{2} - 2050 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[2, 3]$ |
| |
| Discriminant: |
\(-6381708046532608\)
\(\medspace = -\,2^{15}\cdot 41^{7}\)
|
| |
| Root discriminant: | \(94.54\) |
| |
| Galois root discriminant: | $2^{11/4}41^{7/8}\approx 173.38773074713592$ | ||
| Ramified primes: |
\(2\), \(41\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-82}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| This field has no CM subfields. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{7}a^{4}+\frac{3}{7}a^{2}+\frac{1}{7}$, $\frac{1}{7}a^{5}+\frac{3}{7}a^{3}+\frac{1}{7}a$, $\frac{1}{7714}a^{6}-\frac{3}{3857}a^{4}+\frac{1507}{7714}a^{2}+\frac{1385}{3857}$, $\frac{1}{38570}a^{7}+\frac{548}{19285}a^{5}+\frac{4813}{38570}a^{3}-\frac{1921}{19285}a$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
| Ideal class group: | $C_{4}\times C_{4}$, which has order $16$ |
| |
| Narrow class group: | $C_{2}\times C_{4}\times C_{4}$, which has order $32$ |
|
Unit group
| Rank: | $4$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{9}{3857}a^{6}+\frac{71}{551}a^{4}+\frac{7502}{3857}a^{2}+\frac{17767}{3857}$, $\frac{4}{133}a^{6}+\frac{71}{133}a^{4}-\frac{1135}{133}a^{2}-\frac{263}{133}$, $\frac{4}{3857}a^{6}-\frac{24}{3857}a^{4}-\frac{1686}{3857}a^{2}-\frac{491}{3857}$, $\frac{57\cdots 36}{3857}a^{7}-\frac{18\cdots 82}{3857}a^{6}+\frac{29\cdots 93}{3857}a^{5}-\frac{95\cdots 95}{3857}a^{4}+\frac{38\cdots 03}{3857}a^{3}-\frac{17\cdots 63}{551}a^{2}+\frac{11\cdots 94}{3857}a-\frac{36\cdots 93}{3857}$
|
| |
| Regulator: | \( 40907.8145293 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{3}\cdot 40907.8145293 \cdot 16}{2\cdot\sqrt{6381708046532608}}\cr\approx \mathstrut & 4.06468968743 \end{aligned}\]
Galois group
$C_2\wr D_4$ (as 8T35):
| A solvable group of order 128 |
| The 20 conjugacy class representatives for $C_2 \wr C_2\wr C_2$ |
| Character table for $C_2 \wr C_2\wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{41}) \), 4.4.551368.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 8 siblings: | data not computed |
| Degree 16 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| Minimal sibling: | 8.6.1595427011633152.3 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.1.0.1}{1} }^{4}$ | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | R | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.2.3a1.1 | $x^{2} + 2$ | $2$ | $1$ | $3$ | $C_2$ | $$[3]$$ |
| 2.1.2.2a1.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $$[2]$$ | |
| 2.1.4.10a1.1 | $x^{4} + 4 x^{3} + 2$ | $4$ | $1$ | $10$ | $D_{4}$ | $$[2, 3, \frac{7}{2}]$$ | |
|
\(41\)
| 41.1.8.7a1.7 | $x^{8} + 1476$ | $8$ | $1$ | $7$ | $C_8$ | $$[\ ]_{8}$$ |